DOCSLIB.ORG

AN INJECTIVE CHARACTERIZATION OF' PEANO SPACES Burkhard

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
AN INJECTIVE CHARACTERIZATION OF' PEANO SPACES Burkhard Topology and its Applications 11 (1980) 37-46 @ North-Holland Publishing Company AN INJECTIVE CHARACTERIZATION OF’ PEANO SPACES Burkhard HOFFMANN Mathematical Institute, Oxford, England Received 20 November 1978 Revised 26 April 1979 It is shown that a continuous map defined on a closed zero-dimensional sabspace S of a compact space 7’ into a Peano space X can be continuously extended over T or, equivalently, X is an AE(0, OO),and this property precisely characterizesPeano spaces within the class of compact metric spaces. Surjectively, a compact AE(0, ~a)of arbitraryweight is proved to be the continuous image of a Tychonoff cube by a map satisfying the zero-dimensional lifting property. AMS (MOS) Subj. Class. (1970): Primary 54F25,54ClO Peano space zero-dimensional lifting propel :y AE(m, n) 0. Introduction Locally connected, connected metric spaces or Peano spaces were characterized by Hahn and Mazurkiewicz in their celebrated theorem as the continuous images of the unit interval. Although considerable effort has been made in various directions to generalize their theorem to non-met&able spaces (see e.g. [ 111, [ 121 and their references), no satisfactnry solution has yet been fcund. Combining the standard technique of the proof of the Hahn-Mazurkiewicz theorem with the concept of zero-dimensional lifting property or z.d.1.p. we prove that, in the category of compact spaces and continuous maps, Peano spaces are precisely the metrizable absolute extensors for zero- to infinite-dimensional spaces (Theorem 2.2). This result furnishes us with an injective characterization of Peano spaces which seems to be suitable for a generalization to compact spaces of uncountable weight. Theorem 2.1 provides a surjective characterization of this generalized class. For pairs (m, n) of extended integers,, 0 G m G n s 00, a compact space X is defined to be an ahlute extensor for m- to n-dimensional spaces or AE(m, n), if for any embedding C:S + T of compact spaces S, T with dim S G m, dim T 6 n and any continuous map t# : S +X there exists a continuous map $: T -) X such that $0 c = 4. In [4], R. Haydon proved the equivalence of the notions of Dugundji space and AE(0,0). Obviously, a compact space is an absolute retract if and only if it is an AE@, 00). As a hybrid of these two extreme cases, we characterize the class of Peano 37 38 B. Hoffmann / An injective characterization of Peano spaces spaces as the class of metrizable AE(0, a). Furthermore, without restriction to metrizabillity an AE(0,00) is surjectively classified as a continuous image of a Tychonoff cube by a map satisfying the z.d.1.p. (Theorem 2.1). This surjective characterization of AE(0,00) should be contrasted with the characterization given in [S] for AE(0,0) as the continuous image of a Cantor cube be a map satisfying the z.d.1.p. Recall that, as a consequence of Milutin’s lemma, a compact .metrizable space is the continuous image of the Cantor set by a map allowing a regular averaging operator or r.a.o. (see [8], Theorem 5.6). Applying results of a previous paper [S], we deduce here that a Peano space is a continuous image of the unit interval by a map allowing an r.a.o. (Theorem 2.2). This article is devided into two sections. In Section 1 we build up a machinery concerning the concept of z.d.1.p. and culminating in Proposition 1.6 which, in turn, is used as our main tool for the proofs in Section 2. We like to point out that the results and proofs contained in this paper are of a purely topological nature. The reader who is not familiar with the algebraic notion of r.a.o. may choose to substitute it by the topological notion of z.d.1.p. whenever the compact spaces involved are metrizable. For the equivalent of the notions of r.a.o. and z.d.1.p. within the realm of compact metrizable spaces was obtained in a previous article [S]. Notations The closed unit interval of the reals is denoted by I and D is the discrete two-point space (0, 1). We follow the convention of [lo] in identifying a cardinal number with the corresponding initial ordinal. Hence the first two infinite cardinals are indicated by the symbols or)and ml, A countable product of copies of D is homeomorphic to Do. The space R”‘ is frequently identified with the Cantor set, consisting of those elements of I that have a triadic expansion in which the digit 1 does not occur. The dimension function dim indicates, as usual, the covering dimension. Note that dim X .= -1 means X = 0. The weight of a compact space X is the smallest cardinal r such that t&ere exists a base for the topology of X of cardinality T. We can embed X in the Tychonoff cube I’, For a compact space Z denote by C(Z) the Banach space of continuous real- valued functions of 2, equipped Gth the supremum norm. Suppose we are given a continuous surjection 8: X + Y ck compact spaces X, Y. A linear map u: C(X) * C(Y) is said to be a regular aver aging operator or r.a.0, for 8, if u is positive and u( f 00) = f holds for all elements 1; of C( Y). Necessarily, an r.a.0. is of norm 1. 1. The concept of zero-dimensional Ming property The fundamental tool for this piece of research is provided by the notion of zero-dimensional lifting property which we introduced in [S]. For the convenience of the reader we record it here again. B. Hoffmann / An injective characterization of Peano spaces 39 Definition 1.1. Let X, Y be compact spaces and 8: X + Y a continuous surjection. Then 0 satisfies the zero-dimensional lifting property or z.dJ.p., if for any zero- dimensional compact space T and for any continuous map 4: T + Y there exists a continuous lifting a: T + X such that the diagram commutes, i.e. 80 c = 4. Let us start the investigation of the notion of z.d.1.p. by compiling some elementary observations. Trivially, any retraction of compact spaces satisfies the z.d.1.p. and if the image space of a continuous surjection is zero-dimensional the converse holds as well. Any projection of a product of compact spaces onto one of its factors satisfies the z.d.1.p. The notion of z.d.1.p. is stable under finite compositions. Lifting maps coordinatewise immediately yields that the product map (e,)a,A: n xa+ n yk (Xa)acAH(oa(xa))aEA9 aeA aEA satisfies the z.d.l.p., if each coordinate map 6a : Xa + Ya has this property. In [SJ, we proved that for a continuous surjection of compact spaces satisfying the z.d.1.p. entails the existence of an r.a.o. Moreover, it was shown that the converse holds, if the compact spaces involved are metrizable. Hence, in particular, an open continuous surjection of compact metrizable spaces satisfies the z.d.l.p., since it allows an r.a.o. by a result of Pelczyriski [8, p. 651. In the sequel, we shall utilize a lemma which expresses the stability of the notion of z.d.1.p. under the operations of restriction and extension. Its straightforward verification is left to the reader. Lemma 1.2. Let 8: X + Y be a continuous surjection of compact spaces X and Y satisfying the z.d.1.p. Then the following hold. (i) For any closed subset 2 of Y the induced restriction 6(19-‘(Z) : K’(Z) + 2 satisfies the z.d.1.p. (ii) If c : X + 2 is an embedding of X into a compact spti:::e2 and 9: 2 + Y is a continuous extension of 8, i.e. $0 L = 8: then $ satisfies the z.d.1.p. The following result on the notion of z.d.1.p. relies heavily on the fact that we are concerned with the lifting of maps defined on a zero-dimensional domain space. We show that the concept of z.d.1.p. is, by its nature, a local one. o&ion 1.3. Let 8 : X + Y be a continuous surjection of compact spaces Xand Y. Suppose, we are given a family of closed subspaces ( Ya )acA of Ysuch that the union of 40 B. Hoffmann / An injective characterization of Peano spaces the interiors of the sets Ya(a E A) covers Y and a farnil) (Xo)ac~ of closed subspaces of X such that 8 maps Xa onto Y, for all a E A. Let 0,: X, + Y, (a E A) denote the restriction of 8 and assume &, satisfies the z. d.1.p. for all a E A. Then 8 satisfies the a.d.1.p. Proof. Let T be a zero-dimensional compact space and 4 : T + Y a continuous map. Utilizing the z.d.1.p. of the maps & and the zero-dimensionality of the space T we shall construct a continuous lifting a: T + X for 4 by defining it locally. Let Ua denote the interior of Y, for all a E A. By a standard compactness argument there exists a finite subset F of A such that the union U {U, : a E F} coincides with Y. The continuity of 4 entails that the family (&-‘( &))dE~ is a finite, open covering of T.
Load more

Recommended publications
  • D-18 Theˇcech–Stone Compactifications of N and R
    d-18 The Cech–Stoneˇ compactifications of N and R 213 d-18 The Cech–Stoneˇ Compactifications of N and R c It is safe to say that among the Cech–Stoneˇ compactifica- The proof that βN has cardinality 2 employs an in- tions of individual spaces, that of the space N of natural dependent family, which we define on the countable set numbers is the most widely studied. A good candidate for {hn, si: n ∈ N, s ⊆ P(n)}. For every subset x of N put second place is βR. This article highlights some of the most Ix = {hn, si: x ∩ n ∈ s}. Now if F and G are finite dis- striking properties of both compactifications. T S joint subsets of P(N) then x∈F Ix \ y∈G Iy is non-empty – this is what independent means. Sending u ∈ βN to the P( ) ∗ point pu ∈ N 2, defined by pu(x) = 1 iff Ix ∈ u, gives us a 1. Description of βN and N c continuous map from βN onto 2. Thus the existence of an independent family of size c easily implies the well-known The space β (aka βω) appeared anonymously in [10] as an N fact that the Cantor cube c2 is separable; conversely, if D is example of a compact Hausdorff space without non-trivial c dense in 2 then setting Iα = {d ∈ D: dα = 0} defines an converging sequences. The construction went as follows: for ∗ ∈ independent family on D. Any closed subset F of N such every x (0, 1) let 0.a1(x)a2(x)..
    [Show full text]
  • Absolute Measurable Spaces ENCYCLOPEDIAOF MATHEMATICSAND ITS APPLICATIONS
    A bsolute M easurable S paces Absolutemeasurablespaceandabsolutenull spaceareveryold topological notions, developed from descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition aretheaction of thegroup of homeomorphismsof a spaceon Borelmeasures,the Oxtoby–Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem, and recent advances on homeomorphic Borel probability measures on the Cantor space are among the many topics discussed. A brief discussion of set-theoretic results on absolutenull spaceis also given. A four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. The exposition will suit researchers and graduate students of real analysis, set theory and measure theory. Togo Nishiurais Professor Emeritus at Wayne State University, Detroit, andAsso- ciate Fell in Mathematics at Dickinson College, Pennsylvania. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed bel can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=EOM! 63. A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65. K. Engel Sperner Theory 66. D. Cvetkovic, P. Rlinson, and S. Simic Eigenspaces of Graphs 67. F. Bergeron, G. Labelle, and P. Leroux Combinational Species and Tree-Like Structures 68. R. Goodman and N.
    [Show full text]
  • Dense Subgroups of Compact Groups
    Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved Dense subgroups of compact groups W. W. Comfort 1. Introduction The symbol G here denotes the class of all infinite groups, and TG denotes the class of all infinite topological groups which satisfy the T0 separation prop- erty. Each element of TG is, then, a Tychonoff space (i.e., a completely regular, Hausdorff space) [50, 8.4]. We say that G = (G, T ) ∈ TG is totally bounded (some authors prefer the expression pre-compact) if for every U ∈ T \ {∅} there is a finite set F ⊆ G such that G = FU. Our point of departure is the following portion of Weil’s Theorem [77]: Every totally bounded G ∈ TG embeds as a dense topological subgroup of a compact group G; this is unique in the sense that for every compact group Ge containing G densely there is a homeomorphism-and- isomorphism ψ : G Ge fixing G pointwise. As usual, a space is ω-bounded if each of its countable subsets has compact closure. For G = (G, T ) ∈ TG we write G ∈ C [resp., G ∈ Ω; G ∈ CC; G ∈ P; G ∈ TB] if (G, T ) is compact [resp., ω-bounded; countably compact; pseudocompact; totally bounded]. And for X ∈ {C, Ω, CC, P, TB} and G ∈ G we write G ∈ X0 if G admits a group topology T such that (G, T ) ∈ X. The class-theoretic inclusions C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG and C0 ⊆ Ω0 ⊆ CC0 ⊆ P0 ⊆ TB0 ⊆ TG0 = G are easily established. (For P ⊆ TB, see [31].
    [Show full text]
  • Topology Proceedings
    Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 27, No. 2, 2003 Pages 545{558 THE TOPOLOGY OF COMPACT GROUPS SIDNEY A. MORRIS∗ Abstract. This paper is a slightly expanded version of a plenary ad- dresss at the 2002 Summer Topology Conference in Auckland, New Zealand. The aim in this paper is not to survey the vast literature on compact groups, or even the topological group structure of compact groups which is described in considerable detail in the 1998 book \The Structure of Compact Groups: A Primer for the Student { A Handbook for the Expert" by Karl Heinrich Hofmann and this author. Rather, the much more modest purpose in this article is to focus on point-set topology and, in a gentle fashion, describe the topological structure of compact groups, most of which can be extracted or derived from that large book. 1. Fundamental Facts Portions of the material presented here can be found in various papers and books. The book ([3],[4]) by Karl Heinrich Hofmann and the author is a comfortable reference. However, this is not meant to imply that all the results in [3] were first proved by Hofmann and/or Morris, although often the presentation and approach in the book are new. Definition 1.1. Let G be a group with a topology τ. Then G is said to be a topological group if the maps G G given by g g 1 and G G G ! 7! − × ! given by (g1; g2) g1:g2 are continuous.
    [Show full text]
  • A B S T R a C T. We Show That There Is a Scattered Compact Subset K of The
    Bulletin T.CXXXI de l’Acad´emieserbe des sciences et des arts ¡ 2005 Classe des Sciences math´ematiqueset naturelles Sciences math´ematiques, No 30 REPRESENTING TREES AS RELATIVELY COMPACT SUBSETS OF THE FIRST BAIRE CLASS S. TODORCEVIˇ C´ (Presented at the 7th Meeting, held on October 29, 2004) A b s t r a c t. We show that there is a scattered compact subset K of the first Baire class, a Baire space X and a separately continuous mapping f : X £ K ¡! R which is not continuous on any set of the form G £ K, where G is a comeager subset of X. We also show that it is possible to have a scattered compact subset K of the first Baire class which does have the Namioka property though its function space C(K) fails to have an equivalent Fr´echet-differentiable norm and its weak topology fails to be σ-fragmented by the norm. AMS Mathematics Subject Classification (2000): 46B03, 46B05 Key Words: Baire Class-1, Function spaces, Renorming Recall the well-known classical theorem of R.Baire which says that for ev- ery separately continuous real function defined on the unit square [0; 1]£[0; 1] is continuous at every point of a set of the form G£[0; 1] where G is a comea- ger subset of [0; 1]. This result was considerably extended by I.Namioka [12] who showed that the same conclusion can be reached for separately contin- uous functions on products of the form X £ K where X is a Cech-complete space and where K is a compact space.
    [Show full text]
  • TOPOLOGICAL DYNAMICS 1. Introduction Definition 1.1. A
    TOPOLOGICAL DYNAMICS STEFAN GESCHKE 1. Introduction Definition 1.1. A topological semi-group is a topological space S with an associative continuous binary operation ·. A topological semi-group (S; ·) is a topological group if (S; ·) is a group and taking inverses is continuous. Now let S be a topological semi-group and X a topological space. Given a map ' : S ×X ! X, we write sx for '(s; x). ' is a continuous semi-group action if it is continuous and for all s; t 2 S we have s(tx) = (st)x. A dynamical system is a topological semi-group S together with a compact space X and a continuous semi-group action of S on X. X is the phase space of the dynamical system. Sometimes we say that X together with the S-action is an S-flow. Exercise 1.2. Let S be a topological semi-group acting continuously on a compact space X. Show that for all s 2 S the map fs : X ! X; x 7! sx is continuous. Exercise 1.3. Let G be a topological group acting continuously on a compact space X. Show that for all g 2 G the map fg : X ! X; x 7! gx is a homeomorphism. Example 1.4. Let X be a compact space and let f : X ! X be a continuous map. We define a semi-group action of (N; +) on X by letting nx = f n(x) for each n 2 N. This action is continuous with respect to the discrete topology on N. If f : X ! X is a homeomorphism, we can define nx = f n(x) for all nZ and obtain a group action of (Z; +) on X.
    [Show full text]
  • A Product of Compact Normal Spaces Is Normal
    Poorly Separated Infinite Normal Products N. Noble* Abstract From the literature: a product of compact normal spaces is normal; the product of a countably infinite collection of non-trivial spaces is normal if and only if it is countably paracompact and each of its finite sub-products is normal; if all powers of a space X are normal then X is compact – provided in each case that the spaces involved are T1. Here I examine the situation for infinite products not required to be T1 (or regular), extending or generalizing each of these facts. In addition, I prove some related results, give a number of examples, explore some alternative proofs, and close with some speculation regarding potential applications of these findings to category theory and lattice theory. I plan to consider the normality of poorly separated finite products in a companion paper currently in preparation. Summary of principal results Calling a space vacuously normal if it does not contain a pair of nonempty disjoint closed subsets, and using the notation Πα SXα = XS, I show: • A product of vacuously normal spaces is vacuously normal. • A product of compact normal spaces is compact normal. • All powers of X are normal if and only if X is vacuously normal or compact normal. • If X is vacuously normal, Y normal, and the projection from X x Y to Y is closed, then X x Y is normal; the converse holds for Y a T1 space, but not in general. • A countable product with no factor vacuously normal is normal if and only if it is countably paracompact and each of its finite sub-products is normal.
    [Show full text]
  • The Universal Minimal Space for Groups of Homeomorphisms of H
    THE UNIVERSAL MINIMAL SPACE FOR GROUPS OF HOMEOMORPHISMS OF H-HOMOGENEOUS SPACES ELI GLASNER AND YONATAN GUTMAN To Anatoliy Stepin with great respect. Abstract. Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G = Homeo(X), is the space of maximal chains on X introduced in [Usp00]. If X is metrizable then clearly X is homeomorphic to the Cantor set and the result was already known (see [GW03]). However many new examples arise for non-metrizable spaces. These include, among others, the generalized Cantor sets X = {0, 1}κ for non-countable cardinals κ, and the corona or remainder of ω, X = βω \ ω, where βω denotes the Stone-Cˇech compactification of the natural numbers. 1. Introduction The existence and uniqueness of a universal minimal G dynamical system, corre- sponding to a topological group G, is due to Ellis (see [Ell69], for a new short proof see [GL11]). He also showed that for a discrete infinite G this space is never metrizable, arXiv:1106.0121v2 [math.DS] 13 Oct 2011 and the latter statement was generalized to the locally compact non-compact case by Kechris, Pestov and Todorcevic in the appendix to their paper [KPT05]. For Polish groups this is no longer the case and we have such groups with M(G) being trivial (groups with the fixed point property or extremely amenable groups) and groups with 1 1 metrizable, easy to compute M(G), like M(G)= S for the group G = Homeo+(S ) 2010 Mathematics Subject Classification.
    [Show full text]
  • Topological Properties of the Cantor Spaces
    الجـامعــــــــــة اﻹســـــﻼميــة بغــزة The Islamic University of Gaza عمادة البحث العلمي والدراسات العليا Deanship of Research and Graduate Studies كـليــــــــــــــــــــة العـلـــــــــــــــــــــوم Faculty of Science Master of Mathematical Science ماجستيـــــــر العلـــــــوم الـريـاضـيــة Topological Properties of the Cantor Spaces الخصائص التبولوجية لفضاءات كانتور By Ahamed M. Alnajjar Supervised by Prof. Dr. Hisham B. Mahdi Professor of Mathematices A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences. October/2018 إقــــــــــــــرار أنا الموقع أدناه مقدم الرسالة التي تحمل العنوان: Topological Properties of the Cantor Spaces ﺍلﺨﺼائص ﺍلتبولوﺟﻴة لﻔضاءات ﻛانتوﺭ أقر بأن ما اشتملت عليو ىذه الرسالة إنما ىه نتاج جيدي الخاص، باستثناء ما تمت اﻹشارة إليو حيثما ورد، وأن ىذه الرسالة ككل أو أي جزء منيا لم يقدم من قبل اﻻخرين لنيل درجة أو لقب علمي أو بحثي لدى أي مؤسدة تعليمية أو بحثية أخرى. Declaration I understand the nature of plagiarism, and I am aware of the University’s policy on this. The work provided in this thesis, unless otherwise referenced, is the researcher's own work, and has not been submitted by others elsewhere for any other degree or qualification. اسم الطالب: أحمد دمحم خالد النجار Student's name: Ahamed M. Alnajjar التهقيع: :Signature التاريخ: :Date i Abstract ﻓﻲ هذه الرسالة ، نتحدث بشكل رئيسﻲ عن ﻓضاءات كانتور ومكعب كانتور ﻷية مجموعات غير منتهيه ﻓﻲ X ﻓان مكعب كانتور تم تعريفه على انه مجموعات الطاقة من (P(X من X. وهذا الفضاء له تمثيﻼت أخرى ويتكون اﻷساس لهذا الفضاء من كل الفترات التﻲ تكون على شكل [ك, م] حيث ان ك منتهيه و م غير منتهيه وحيث تم استخدام هذا اﻷساس بفعاليه ودرسنا خصائص الفضاءات الكانتور و مكعب كانتور وأنواع مختلفه من مكعب كانتور ,وأثبتنا أن الفضاء كانتور هو الفضاء الوحيد (ما يصل إلى التماثل) التﻲ تجمع بين خصائص الفضاء المنفصل ، والكمال ، والمدمج ، والمتر المترابط تما ًما وأثبتنا أن µ تكون تبولوجﻲ ألكساندر وف ﻓإنها تكون مغلقه ﻓﻲ 푿 .
    [Show full text]
  • F-9 Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces
    hart v.2003/06/18 Prn:20/06/2003; 9:30 F:hartf10.tex; VTEX/J p. 1 334 Section F: Special properties f-9 Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces A space X is called zero-dimensional if it is nonempty theorems give us homogeneity for free. (This phenomenon is and has a base consisting of clopen sets, i.e., if for every not uncommon in topology.) point x ∈ X and for every neighbourhood U of x there The following example of a zero-dimensional compact exists a clopen subset C ⊆ X such that x ∈ C ⊆ U.It space is of particular interest. From I =[0, 1] remove the 1 2 is clear that a nonempty subspace of a zero-dimensional interval ( /3, /3), i.e., the ‘middle-third’ interval. From the space is again zero-dimensional and that products of zero- remaining two intervals, again remove their ‘middle-thirds’, dimensional spaces are zero-dimensional. and continue in this way infinitely often. What remains of I We say that a space X is totally disconnected if for all at the end of this process is called the Cantor middle-third distinct points x,y in X there exists a clopen set C in set, C. A space homeomorphic to C is called a Cantor set. ∈ ∈ X such that x C but y/C. It is clear that every zero- It is easy to see that C is the subspace of I consisting of dimensional space is totally disconnected. The question nat- all points that have a triadic expansion in which the digit 1 urally arises whether every totally disconnected space is does not occur, i.e., the set zero-dimensional.
    [Show full text]
  • A Note on the $ Luc-$ Compactification of Locally Compact Groups
    A NOTE ON THE LUC-COMPACTIFICATION OF LOCALLY COMPACT GROUPS SAFOURA JAFAR-ZADEH Abstract. Let G be a locally compact non-compact topological group. We show that Gluc (G∗) is an F-space if and only if G is a discrete group. 1. Introduction Let G be a Hausdorff locally compact group and Cb(G) denote the Banach space of bounded continuous functions on G with the uniform norm. An element f in Cb(G) is called left uniformly continuous if g 7→ lgf, where lgf(x)= f(gx) is a continuous map from G to Cb(G). Let LUC(G) denote the Banach space of left uniformly continuous functions with the uniform norm. It can be seen that Cc(G) the space of continuous compactly supported functions on G is a subspace of LUC(G). The dual space of LUC(G), LUC(G)∗, with the following Arens-type product forms a Banach algebra: < m.n, f >=< m, nf > and < nf, g >=< n,lg(f) >, where m, n ∈ LUC(G)∗, f ∈ LUC(G) and g ∈ G. In fact this product is an extension to LUC(G)∗ of the convolution product on M(G). The luc-compactification of a locally compact group G, denoted by Gluc can be regarded as a generalization to locally compact groups, of the familiar Stone-Cech compactification for discrete groups. As a topological space, Gluc is the Gelfand spectrum of the unital commutative C∗−algebra of left uniformly continuous func- tions on G, i.e., LUC(G) = C(Gluc). The restriction of the product on LUC(G)∗ arXiv:1401.4775v1 [math.FA] 20 Jan 2014 to the subset Gluc of LUC(G)∗ turns Gluc into a semigroup.
    [Show full text]
  • Author's Review of His Research, Achievements and Publications
    Author's review of his research, achievements and publications 1. Name: Szymon Zeberski_ 2. Obtained diplomas and academic degrees: • M.Sc. in Mathematics, Mathematical Institute, University of Wroc law, 2000 • Ph.D. in Mathematics, Mathematical Institute, University of Wroc law, 2004 3. Employment in scientific institutions: • Assistant Lecturer, Mathematical Institute, University of Wroc law, 2004{2005, • Assistant Professor, Mathematical Institute, University of Wroc law, 2005{2007, • Assistant Professor, Institute of Mathematics and Computer Science, Wroc law University of Technology, 2007{2014, • Assistant Professor, Department of Computer Science, Faculty of Fundamental Problems of Technology, Wroc law University of Technology, since 2014. 4. Achievements in the form of series of publications under the title Nonmeasurability in Polish spaces with respect to Article 16 Paragraph 2 of the Act of 14 March 2003 on Academic Degrees and Title and on Degrees and Title in the Field of Art. List of publications included in the achievement mentioned above [H1] Sz. Zeberski_ , On weakly measurable functions, Bulletin of the Polish Academy of Sciences, Mathematics, 53 (2005), 421-428, [H2] Sz. Zeberski_ , On completely nonmeasurable unions, Mathematical Logic Quar- terly, 53 (1) (2007), 38-42, [H3] R. Ra lowski, Sz. Zeberski_ , Complete nonmeasurability in regular families, Houston Journal of Mathematics, 34 (3) (2008), 773-780, [H4] R. Ra lowski, Sz. Zeberski_ , On nonmeasurable images, Czechoslovak Mathemati- cal Journal, 60 (135) (2010), 424-434, [H5] R. Ra lowski, Sz. Zeberski_ , Completely nonmeasurable unions, Central European Journal of Mathematics, 8 (4) (2010), 683-687, [H6] Sz. Zeberski_ , Inscribing nonmeasurable sets, Archives for Mathematical Logic, 50 (3) (2011), 423-430, [H7] R.
    [Show full text]